Continuous extensions of functions defined on subsets of products

A subset Y of a space X is G δ -dense if it intersects every nonempty G δ -set. The G δ -closure of Y in X is the largest subspace of X in which Y is G δ -dense. ace X has a regular G δ -diagonal if the diagonal of X is the intersection of countably many regular-closed subsets of X × X . er now these results: (a) (N. Noble, 1972 [18]) every G δ -dense subspace in a product of separable metric spaces is C-embedded; (b) (M. Ulmer, 1970 [22], 1973 [23]) every Σ-product in a product of first-countable spaces is C-embedded; (c) (R. Pol and E. Pol, 1976 [20], also A.V. Arhangelʼskiĭ, 2000 [3]; as corollaries of more general theorems), every dense subset of a product of completely regular, first-countable spaces is C-embedded in its G δ -closure. esent authorsʼ Theorem 3.10 concerns the continuous extension of functions defined on subsets of product spaces with the κ-box topology. Here is the case κ = ω of Theorem 3.10, which simultaneously generalizes the above-mentioned results. Theorem X i : i ∈ I } be a set of T 1 -spaces, and let Y be dense in an open subspace of X I : = ∏ i ∈ I X i . If χ ( q i , X i ) ⩽ ω for every i ∈ I and every q in the G δ -closure of Y in X I , then for every regular space Z with a regular G δ -diagonal, every continuous function f : Y → Z extends continuously over the G δ -closure of Y in X I . xamples are cited to show that the hypothesis χ ( q i , X i ) ⩽ ω cannot be replaced by the weaker hypothesis ψ ( q i , X i ) ⩽ ω .